By Charles Kittel

A contemporary presentation of theoretical reliable kingdom physics that builds at once upon Kittel's Introduction to sturdy country Physics. Treats phonon, electron, and magnon fields, culminating within the BCS conception of superconductivity. Considers Fermi surfaces and electron wave services and develops the crowd theoretical description of Brillouin zones. Applies correlation features to time-dependent results in solids, with an advent to Green's services. With a hundred and ten difficulties, the textual content is well-suited for the study room or for self-instruction.

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Over the basis of Q(q±112) given by powers of q112, we may write J(L; q) as a string of integers, which is easily encoded as bit string. Hence with encodings the computation of J(L; q) is a map J(L; q) : {0, 1}* --+ {0, 1}* Can we compute J(L; q) efficiently? The computation can be either quantum me­ chanical or classical. Moreover rather than the exact answer we can ask for an approximation according to various approximation schemes. Therefore there are many ways to formulate the computation of the Jones polynomial as a computing problem.

2) A fusion rule on a label set L is a binary operation ® : L L --+ N L , where N L is the set of all maps from L to N = {0, 1, 2, . . } satisfying the following conditions. First we introduce some notation. Given a, b E L, we will write formally a ® b = E9 N�b c where N�b = (a ® b) (c) . When no confusion arises, we write a ® b simply as ab, so a2 = a ® a. Then the conditions on ® are: for all a, b, c, d E L, (i) (a ® b) ® c = a ® (b ® c) , i. , L xE L N�b N�c = LxE L Nbc N�x (ii) N�1 = N'fa bca (iii) N};b = Nla iSba We say a triple of labels (a, b, c) is admissible if N�b -1- 0.

We have already seen two different normalizations of the Jones representation at r = 4 or level k = 2: the Jones normalization and the Kauffman bracket nor­ malization. There is also a third normalization related to the "(-matrices: " � pA_ (ui) = e •"�'+1"�' = 1 ( 1 + 'Yi +l 'Yi) · J2 The orders of p{(ui) , p}(ui), and PA(ui) are 4, 8, and 16 respectively. Although all three normalizations have the same projective image, their linear images are in general different. _ 1 , the image is E;;� 1 [FRW] .

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